We report a systematic investigation of the mechanical properties of fibroblast cells using a novel cell monolayer rheology (CMR) technique. The new technique provides quantitative rheological parameters averaged over ~106 cells making the experiments highly reproducible. Using this method, we are able to explore a broad range of cell responses not accessible using other present day techniques. We perform harmonic oscillation experiments and step shear or step stress experiments to reveal different viscoelastic regimes. The evolution of the live cells under externally imposed cyclic loading and unloading is also studied. Remarkably, the initially nonlinear response becomes linear at long timescales as well as at large amplitudes. Within the explored rates, nonlinear behaviour is only revealed by the effect of a nonzero average stress on the response to small, fast deformations. When the cell cytoskeletal crosslinks are made permanent using a fixing agent, the large amplitude linear response disappears and the cells exhibit a stress stiffening response instead. This result shows that the dynamic nature of the cross-links and/or filaments is responsible for the linear stress-strain response seen under large deformations. We rule out the involvement of myosin motors in this using the inhibitor drug blebbistatin. These experiments provide a broad framework for understanding the mechanical responses of the cortical actin cytoskeleton of fibroblasts to different imposed mechanical stimuli.